A Characterization of Borel Measures which Induce Lipschitz-Free Space Elements
| Autor: | Raad, Lucas Maciel |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | We will solve a problem by Aliaga about Lipschitz-Free spaces (denoted by $\mathcal F(M)$): $$ \text{Does every Borel measure $\mu$ on a complete metric space $M$ such that $\int d(m,0) d |\mu|(m)< \infty$ induce a weak$^*$ continuous functional $\mathcal L\mu \in \mathcal F(M)$ by the mapping $\mathcal L\mu(f)=\int f d \mu$?}$$ In particular, we will show a characterization of the measures such that $\mathcal L\mu \in \mathcal F(M)$, which indeed implies inner-regularity for complete metric spaces, and we will prove that every Borel measure on $M$ induces an element of $\mathcal F(M)$ if and only if the weight of $M$ is strictly less than the least real-valued measurable cardinal, and thus the existence of a metric space on which there is a measure $\mu$ such that $\mathcal L\mu \in \mathcal F(M)^{**} \setminus \mathcal F(M)$ cannot be proven in ZFC. Comment: 12 pages. For the original problem see Problem 4.2 on https://www.researchgate.net/publication/344282833_Geometry_and_structure_of_Lipschitz-free_spaces_and_their_biduals or Question 2 on arXiv:2009.07663 |
| Databáze: | arXiv |
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